   Chapter 3.1, Problem 30E ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

#### Solutions

Chapter
Section ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
14 views

# In Exercises 27 to 32, use SSS, SAS, ASA, or AAS to prove that the triangles are congruent. Given: P N ¯ bisects M Q ¯ ∠ M and ∠ Q are right angles Prove: Δ P Q R ≅ Δ N M R To determine

To prove:

The triangle ΔPQR is congruent to the triangle ΔNMR.

Explanation

Given:

The triangles ΔPQR and ΔNMR are shown in the figure below,

Figure (1)

From the figure (1),

PN¯ bisects MQ¯

And,

M and Q are right angles

Property used:

If two lines intersect, then the vertical angles formed are congruent.

Approach:

Consider the triangles,

ΔPQR and ΔNMR

From the given data,

The angles M and Q are right angles in triangles ΔPQR and ΔNMR.

Since, the angle M is equal to the angle Q, then

MQ

And the line PN¯ bisect the line MQ¯ in two equal lines MR¯ and RQ¯.

Since the lines MR¯ and RQ¯ are equal, then

MR¯RQ¯

Since, the line PN¯ bisect the line MQ¯, then there is vertical angles PRQ and MRN are formed

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